Problem 23 - Calculus Early Transcendentals: Pearson New International Edition

Question

Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$\lim _{x \rightarrow \infty} x^{1 / x}$$

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Identify the Form

To apply l'Hôpital's rule, we first need to determine if the limit is an indeterminate form. The expression given is \( \lim_{x \to \infty} x^{1/x} \). As \( x \to \infty \), \( x^{1/x} \) approaches \( 1 \) since the exponent \( 1/x \to 0 \). This is an indeterminate form of the type \( 1^0 \). However, this is not strictly of the form necessary for l'Hôpital's rule, which usually applies to forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). Thus, we'll transform it appropriately.

Step 2: Transform using Logarithms

Instead of applying l'Hôpital's rule directly, let's take the natural logarithm of the expression to transform it. Consider \( y = x^{1/x} \). Take the logarithm on both sides: \( \ln y = \frac{1}{x} \ln x \). Now find \( \lim_{x \to \infty} \ln y = \lim_{x \to \infty} \frac{\ln x}{x} \), which is a \( \frac{\infty}{\infty} \) indeterminate form, allowing for l'Hôpital's rule to be applied.

Step 3: Apply l'Hôpital's Rule

To compute \( \lim_{x \to \infty} \frac{\ln x}{x} \) using l'Hôpital's rule, differentiate the numerator and denominator separately:- The derivative of \( \ln x \) is \( \frac{1}{x} \).- The derivative of \( x \) is \( 1 \).Apply l'Hôpital's Rule:\[ \lim_{x \to \infty} \frac{\ln x}{x} = \lim_{x \to \infty} \frac{1/x}{1} = \lim_{x \to \infty} \frac{1}{x} = 0. \]

Step 4: Exponentiate to Find the Original Limit

Since \( \ln y = 0 \), we exponentiate both sides to solve for \( y \):\[ y = e^{\ln y} = e^0 = 1. \]Thus, \( \lim_{x \to \infty} x^{1/x} = 1. \)

Key Concepts

l'Hôpital's Rule in Calculus

l'Hôpital's Rule is a powerful technique in calculus, used to solve limits that result in indeterminate forms. It states that if you have a limit in the form of \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), you can differentiate the numerator and the denominator separately to evaluate the limit. This process can sometimes need repeating if the result is still indeterminate.

Here's a simple overview of when and how to apply l'Hôpital's Rule:

• Check if the limit results in \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).
• Differentiating both the numerator and denominator.
• Re-evaluate the limit with the new expression.
• If an indeterminate form arises again, repeat the differentiation.

Remember, l'Hôpital's Rule is applicable only when direct substitution yields these specific forms.

Understanding Indeterminate Forms

Indeterminate forms occur in calculus when evaluating a limit doesn't initially lead to a definite number. Various expressions can take these forms, such as \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), \( 1^\infty \), \( 0^0 \), \( \infty^0 \), and others.

These forms are essential because they indicate a need for further evaluation or transformation. For example, when approaching the limit \( \lim _{x \rightarrow \infty} x^{1/x} \), we recognize the "\( 1^0 \)" form, which requires a transformation before applying any rules.

Identifying the type of indeterminate form guides you on the best method to solve the problem, whether it's through algebraic manipulation, transformation, or calculus techniques like l'Hôpital's Rule.

The Role of Limits in Calculus

Limits are a foundational concept in calculus, describing the behavior of functions as inputs approach a particular value. They are essential for defining derivatives and integrals.

To find a limit, you need to evaluate how a function behaves as it gets closer to a specific point, which might be a number or infinite. The method can vary depending on the situation, such as using direct substitution, algebraic manipulation, or advanced techniques like l'Hôpital's Rule.

Understanding limits are crucial because they help describe continuity, define slopes of curves, and solve problems involving change over time.

Exploring the Natural Logarithm

The natural logarithm, denoted as \( \ln \), is the logarithm to the base \( e \), where \( e \approx 2.71828 \). It is a significant function in calculus because it simplifies various expressions and is integral in digital growth models and compound interest.

Natural logarithms possess properties that make them useful in calculus:

• \( \ln(ab) = \ln(a) + \ln(b) \)
• \( \ln(a^b) = b\ln(a) \)
• \( \ln(1) = 0 \)

In problems like \( x^{1/x} \), taking the natural logarithm transforms the expression to one that can be more easily evaluated with calculus techniques. This is why we use \( \ln \) to manipulate and find limits when the initial expression takes on an indeterminate form.